The men are understood to have been working in a cherry picker when they were engulfed in flames following an explosion.
A John F Hunt team is helping demolish old ammonia washers at the former SSI site following its closure in 2015.
One of the victims worked for John F Hunt and the other for Nationwide Platforms.
A John F Hunt company statement said: “We sincerely regret to confirm that a serious incident occurred at our SSI site in Teesside yesterday.
“Tragically this has resulted in two fatalities and we are currently engaging with the HSE, client and emergency services to thoroughly investigate and ascertain the circumstances surrounding this heart-rending event.
“Our heartfelt thoughts at this time are with the family, friends and colleagues of the two men and we would like to express how deeply sorry we are for their loss and the anguish they will be experiencing.”
Nationwide Platforms said: “It is with deep regret and shock that we confirm that a much-loved Nationwide Platforms employee died as a result of yesterday’s explosion at the former SSI steelworks site in Redcar.”
Mark Keily, Health, Safety and Environmental Director, Nationwide Platforms, said: “Our thoughts are with the victims’ families, friends and colleagues at this awful time.
“We are in close contact with our employee’s family to ensure they receive every bit of support and guidance possible.
“Meanwhile Nationwide Platforms is, among others, assisting the authorities with their enquiries. At this early stage, we can’t comment further.
“As a leading powered access rental provider, we take our responsibilities for health and safety extremely seriously.”